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For clarity: a distance function on a space $X$ is a function $d: X \times X \to \mathbb{R}$ satisfying the axioms for a metric (e.g. triangle inequality). A metric tensor $g$ assigns to each point $p \in X$ a function $g_p: T_p \times T_p \to \mathbb{R}$, where $T_p$ is the tangent space to $p$.

My question: is it possible to derive a metric tensor from a distance function, and vice versa?

For example, in Cartesian coordinates the Euclidean distance corresponds to the metric tensor diag(1, 1) (in two dimensions). But is there also a tensor for the taxicab distance, or for the British Rail distance?

More specifically, does there always exist a tensor such that the smallest curve length between two points is equal to the distance between them according to a given distance function?

Vice versa, for any given metric tensor, is there a distance function satisfying the axioms for a metric such that the distance between two points is equal to the smallest curve length between them?

I hope the question is clear! Thanks in advance for your answers.

Caspar201
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    Are you talking about a Riemannian metric $g$ on $T_pX$, which is a tensor, or a norm $g$ on $T_pX$, which is not a tensor. In particular, the taxicab distance in Cartesian coordinates cannot be represented by a tensor. – Deane Mar 20 '22 at 16:32
  • Hi Deane! Thanks for your answer. I was thinking of a Riemannian metric. Could you please explain to me why the taxicab distance cannot be represented by a tensor? – Caspar201 Mar 20 '22 at 16:41
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    If a distance on $\mathbb{R}^n$ can be represented by a tensor, then, by the properties of distance, the tensor has to be positive definite and symmetric. That's equivalent to saying that it is an inner product, and the unit ball is an ellipsoid. The unit ball of the taxicab distance is not. – Deane Mar 20 '22 at 16:50
  • How much Riemannian geometry have you studied? The second question is presented in most textbooks, and the first is straightforward using the exponential map. – Deane Mar 20 '22 at 16:51
  • I have a physics background, so my knowledge of Riemannian geometry is more geared towards applications in GR. If it's not too much of a burden on your time, could you please explain how you would answer the first question using the exponential map? Thanks! – Caspar201 Mar 20 '22 at 17:05
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    This seems to be a duplicate of this question. The precise answer depends on what you mean by "space" (in particular, is $X$ a general metric space or a smooth manifold?), but both cases are discussed there. – Kajelad Mar 20 '22 at 23:14
  • Thanks Kajelad, I think you are right that answers my question! – Caspar201 Mar 21 '22 at 02:40
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    Here's a variant of the question: Suppose you know that $d$ is the distance function of a Riemannian metric and want to recover the Riemannian metric from $d$. The answer is that at each point $p$, with respect to local coordinates, $$g_{ij}= \frac{1}{2}\partial_{ij}^2(d^2)$$ – Deane Mar 21 '22 at 15:42
  • @Deane thank you!!! In the other post the people say no and why, which is fine, but the don't answer how you would do it in a case where you can. Finally someone that answers the question.... – Alberto Rolandi Apr 26 '22 at 10:03

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